Increasing paths in regular trees

We consider a regular n-ary tree of height h, for which every vertex except the root is labelled with an independent and identically distributed continuous random variable. Taking motivation from a question in evolutionary biology, we consider the number of paths from the root to a leaf along vertices with increasing labels. We show that if α=n/h is fixed and α>1/e, the probability that there exists such a path converges to 1 as h→∞. This complements a previously known result that the probability converges to 0 if α≤1/e.